The embodiments herein generally relate to printing devices and, more particularly, to an optimized limit gain component for a dispense time accumulator of a toner concentration control for dispensing toner in a developer housing of a printing engine.
In general, all printing engines have some form of “Toner Concentration (TC) Control” to maintain the desired amount of toner in a developer housing, which is usually achieved by means of dispensing toner into the developer housing by actuation of the dispense motor to maintain the desired image quality metrics. For cost savings, it is desirable to use DC motors instead of servo or stepper motors. To enable this, the concept of a dispense time accumulator/buffer is commonly used to enable decoupling of controls and actuation.
Use of this concept moves the dispense model away from simple classic continuous or discrete time-invariant systems, making it more complicated. However, one cannot directly apply the commonly used anti-windup architecture (which is directed toward the Integral (I) part of a Proportional-Integral (PI) control loop) that is applied only to synchronous, coupled systems. Therefore, the embodiments herein implement an equivalent anti-windup component in an accumulator/buffer that is decoupled and asynchronously operating from any control loop.
Since all actuators have physical limitations, a DC motor in a developer housing having a limited and finite velocity may have severe consequences for a PI (Proportional, Integral) controller, where integral action is an unstable mode. While this does not cause any difficulties when the loop is closed, a feedback loop will be broken when an actuator saturates (because the output of the saturating element is not influenced by its input). The unstable mode in the controller may then drift to very large values. When the actuator desaturates, it may take a long time for the system to recover, or it may also oscillate several times between high and low before the system recovers.
Integrator windup occurs due to the saturation in the actuator where the control signal saturates immediately when the step is applied. The control signal then remains in saturation level and the feedback is broken and the integral part continues to increase because the error is positive. The integral part starts to decrease when the process output has become larger than the setpoint, but the process output remains saturated because of the large integral part. Slowly, the process output decreases towards the setpoint. The net effect of this process is a large overshoot called, “integrator windup.”
The controller operates linearly only if the process output is in the range of the minimum and maximum output, and the controller output saturates when the process output is outside this range. A controller models in parallel form with the anti-windup scheme having extra proportional band variables equal to the minimum and maximum output based on the integral part. Thus, integrator windup can be avoided by making sure that the integral is kept to a proper value when the actuator saturates so that the controller is ready to resume action, as soon as the control error changes.